mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.Int.WithZero

Goal state before:
pp: e : ℝ≥0
he : 1 < e
⊢ StrictMono ⇑(toNNReal ⋯)

Tactic:
intro x

Goal state after:
e : ℝ≥0
he : 1 < e
x : WithZero (Multiplicative ℤ)
⊢ ∀ ⦃b : WithZero (Multiplicative ℤ)⦄, x < b → (toNNReal ⋯) x < (toNNReal ⋯) b
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3
⊢ W.dblX P / W.dblZ P ^ 2 =
    W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
      (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3))

Tactic:
by
  rw [dblX, toAffine_slope_of_eq hPz hQz hx hy, dblZ, ← (X_eq_iff hPz hQz).mp hx,
    toAffine_addX_of_eq hPz <| sub_ne_zero.mpr <| Y_ne_negY_of_Y_ne' hP hQ hx hy]

Goal state after:
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
⊢ ↑(B i) ⊆ (fun x ↦ -x) ⁻¹' ↑(B i)

Tactic:
intro x

Goal state after:
case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
x : A
⊢ x ∈ ↑(B i) → x ∈ (fun x ↦ -x) ⁻¹' ↑(B i)
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: case h.right
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i : ι
⊢ ↑(B i) ⊆ (fun x ↦ -x) ⁻¹' ↑(B i)

Tactic:
intro x

Goal state after:
case h.right
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i : ι
x : M
⊢ x ∈ ↑(B i) → x ∈ (fun x ↦ -x) ⁻¹' ↑(B i)
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
⊢ ∀ (x₀ : R) {U : Set M}, U ∈ {U | ∃ i, U = ↑(B i)} → ∃ V ∈ {U | ∃ i, U = ↑(B i)}, V ⊆ (fun x ↦ x₀ • x) ⁻¹' U

Tactic:
by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro m
      exact (B i).smul_mem _

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
⊢ ∀ (x y z w : R) (t s u v : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        z * ↑u ≤ᵥ w * ↑v →
          w * ↑v ≤ᵥ z * ↑u → ValueGroupWithZero.mk (x * z) (s * v) = ValueGroupWithZero.mk (y * w) (t * u)

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y y' z x : R
⊢ ∀ (y z w : R) (t s u v : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        z * ↑u ≤ᵥ w * ↑v →
          w * ↑v ≤ᵥ z * ↑u → ValueGroupWithZero.mk (x * z) (s * v) = ValueGroupWithZero.mk (y * w) (t * u)
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
γ : (ofClass v).ValueGroup₀ˣ
γx : Γ₀ˣ
Hx : v x = ↑γx
⊢ (restrict₀ (ofClass v)) x ≠ 0

Tactic:
simp [restrict₀_apply]

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
γ : (ofClass v).ValueGroup₀ˣ
γx : Γ₀ˣ
Hx : v x = ↑γx
⊢ ¬(if h : v x = 0 then 0 else ↑⟨Units.mk0 (v x) ⋯, ⋯⟩) = 0
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
n : ℕ
⊢ ∀ (x y : R) (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s → y * ↑s ≤ᵥ x * ↑t → ValueGroupWithZero.mk (x ^ n) (s ^ n) = ValueGroupWithZero.mk (y ^ n) (t ^ n)

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y y' z : R
n : ℕ
x : R
⊢ ∀ (y : R) (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s → y * ↑s ≤ᵥ x * ↑t → ValueGroupWithZero.mk (x ^ n) (s ^ n) = ValueGroupWithZero.mk (y ^ n) (t ^ n)
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
⊢ ∀ (x y z w : R) (t s u v : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s → y * ↑s ≤ᵥ x * ↑t → z * ↑u ≤ᵥ w * ↑v → w * ↑v ≤ᵥ z * ↑u → (x * ↑v ≤ᵥ z * ↑s) = (y * ↑u ≤ᵥ w * ↑t)

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y y' z x : R
⊢ ∀ (y z w : R) (t s u v : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s → y * ↑s ≤ᵥ x * ↑t → z * ↑u ≤ᵥ w * ↑v → w * ↑v ≤ᵥ z * ↑u → (x * ↑v ≤ᵥ z * ↑s) = (y * ↑u ≤ᵥ w * ↑t)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Jacobian R
f : R →+* S
P : Fin 3 → R
⊢ (W'.map f).neg (⇑f ∘ P) = ⇑f ∘ W'.neg P

Tactic:
simp only [neg, map_negY, comp_fin3]
  map_simp

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x : R
s : ↥(posSubmonoid R)
⊢ 0 < ValueGroupWithZero.mk x s ↔ 0 <ᵥ x

Tactic:
rw [← mk_zero 1]

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x : R
s : ↥(posSubmonoid R)
⊢ ValueGroupWithZero.mk 0 1 < ValueGroupWithZero.mk x s ↔ 0 <ᵥ x
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
⊢ ∀ (x y : R) (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        (if h : x ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑s ⟨x, h⟩) =
          if h : y ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑t ⟨y, h⟩

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y y' z x : R
⊢ ∀ (y : R) (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        (if h : x ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑s ⟨x, h⟩) =
          if h : y ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑t ⟨y, h⟩
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case inr
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ ∀ x ∈ {x | v.restrict x < r}, {x | v.restrict x < r} ∈ 𝓝 x

Tactic:
intro x

Goal state after:
case inr
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
x : R
⊢ x ∈ {x | v.restrict x < r} → {x | v.restrict x < r} ∈ 𝓝 x
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₁ : Type u_3
Γ₂ : Type u_4
inst✝³ : LinearOrderedCommMonoidWithZero Γ₁
inst✝² : LinearOrderedCommMonoidWithZero Γ₂
v₁ : Valuation R Γ₁
v₂ : Valuation R Γ₂
inst✝¹ : v₁.Compatible
inst✝ : v₂.Compatible
⊢ v₁.IsEquiv v₂

Tactic:
intro x

Goal state after:
R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₁ : Type u_3
Γ₂ : Type u_4
inst✝³ : LinearOrderedCommMonoidWithZero Γ₁
inst✝² : LinearOrderedCommMonoidWithZero Γ₂
v₁ : Valuation R Γ₁
v₂ : Valuation R Γ₂
inst✝¹ : v₁.Compatible
inst✝ : v₂.Compatible
x : R
⊢ ∀ (s : R), v₁ x ≤ v₁ s ↔ v₂ x ≤ v₂ s
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ ∀ x ∈ {x | v.restrict x ≤ r}, {x | v.restrict x ≤ r} ∈ 𝓝 x

Tactic:
intro x

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
x : R
⊢ x ∈ {x | v.restrict x ≤ r} → {x | v.restrict x ≤ r} ∈ 𝓝 x
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
⊢ ∀ x ∈ {x | v.restrict x ≤ r}ᶜ, {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x

Tactic:
intro x

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
x : R
⊢ x ∈ {x | v.restrict x ≤ r}ᶜ → {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case neg
R : Type u_1
inst✝³ : Semiring R
inst✝² : ValuativeRel R
inst✝¹ : IsDiscrete R
inst✝ : IsNontrivial R
a : ValueGroupWithZero R
ha : 0 < a
⊢ (uniformizer R)⁻¹ ≤ a ↔ 1 < a

Tactic:
inv_le_comm₀ uniformizer_pos ha,

Goal state after:
case neg
R : Type u_1
inst✝³ : Semiring R
inst✝² : ValuativeRel R
inst✝¹ : IsDiscrete R
inst✝ : IsNontrivial R
a : ValueGroupWithZero R
ha : 0 < a
⊢ a⁻¹ ≤ uniformizer R ↔ 1 < a
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
⊢ ∀ (x y : R) (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        (restrict₀ (ofClass v)) x / (restrict₀ (ofClass v)) ↑s = (restrict₀ (ofClass v)) y / (restrict₀ (ofClass v)) ↑t

Tactic:
intro x

Goal state after:
R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
x : R
⊢ ∀ (y : R) (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        (restrict₀ (ofClass v)) x / (restrict₀ (ofClass v)) ↑s = (restrict₀ (ofClass v)) y / (restrict₀ (ofClass v)) ↑t
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: case inr
R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ ∀ x ∈ {x | v.restrict x < r}, {x | v.restrict x < r} ∈ 𝓝 x

Tactic:
intro x

Goal state after:
case inr
R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
x : R
⊢ x ∈ {x | v.restrict x < r} → {x | v.restrict x < r} ∈ 𝓝 x
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ ∀ x ∈ {x | v.restrict x ≤ r}, {x | v.restrict x ≤ r} ∈ 𝓝 x

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
x : R
⊢ x ∈ {x | v.restrict x ≤ r} → {x | v.restrict x ≤ r} ∈ 𝓝 x
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
⊢ ∀ x ∈ {x | v.restrict x ≤ r}ᶜ, {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
x : R
⊢ x ∈ {x | v.restrict x ≤ r}ᶜ → {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
x✝¹ x✝ : ValueGroupWithZero R
⊢ ∀ (a b a_1 : R) (b_1 : 0 <ᵥ a_1) (a_2 : R) (b_2 : 0 <ᵥ a_2),
    ((if h : (ofClass v) b = 0 then 0
        else
          if h : (ofClass v) a = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) a) ⋯, ⋯⟩ * ↑⟨Units.mk0 ((ofClass v) b) ⋯, ⋯⟩) /
        if h : (ofClass v) a_2 = 0 then 0
        else
          if h : (ofClass v) a_1 = 0 then 0
          else ↑⟨Units.mk0 ((ofClass v) a_1) ⋯, ⋯⟩ * ↑⟨Units.mk0 ((ofClass v) a_2) ⋯, ⋯⟩) =
      ((if h : (ofClass v) a = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) a) h, ⋯⟩) /
          if h : (ofClass v) a_1 = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) a_1) ⋯, ⋯⟩) *
        ((if h : (ofClass v) b = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) b) h, ⋯⟩) /
          if h : (ofClass v) a_2 = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) a_2) ⋯, ⋯⟩)

Tactic:
intro x

Goal state after:
R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
x✝¹ x✝ : ValueGroupWithZero R
x : R
⊢ ∀ (b a : R) (b_1 : 0 <ᵥ a) (a_1 : R) (b_2 : 0 <ᵥ a_1),
    ((if h : (ofClass v) b = 0 then 0
        else
          if h : (ofClass v) x = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) x) ⋯, ⋯⟩ * ↑⟨Units.mk0 ((ofClass v) b) ⋯, ⋯⟩) /
        if h : (ofClass v) a_1 = 0 then 0
        else
          if h : (ofClass v) a = 0 then 0
          else ↑⟨Units.mk0 ((ofClass v) a) ⋯, ⋯⟩ * ↑⟨Units.mk0 ((ofClass v) a_1) ⋯, ⋯⟩) =
      ((if h : (ofClass v) x = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) x) h, ⋯⟩) /
          if h : (ofClass v) a = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) a) ⋯, ⋯⟩) *
        ((if h : (ofClass v) b = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) b) h, ⋯⟩) /
          if h : (ofClass v) a_1 = 0 then 0 else ↑⟨Units.mk0 ((ofClass v) a_1) ⋯, ⋯⟩)
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case mk
R : Type u_2
Γ : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝² : v.Compatible
Γ' : Type u_4
inst✝¹ : LinearOrderedCommGroupWithZero Γ'
w : Valuation R Γ'
inst✝ : w.Compatible
h : w.IsEquiv v
r : R
s : ↥(posSubmonoid R)
⊢ h.orderMonoidIso
      (ValueGroupWithZero.lift (fun r s ↦ (restrict₀ (ofClass w)) r / (restrict₀ (ofClass w)) ↑s) ⋯
        (ValueGroupWithZero.mk r s)) =
    ValueGroupWithZero.lift (fun r s ↦ (restrict₀ (ofClass v)) r / (restrict₀ (ofClass v)) ↑s) ⋯
      (ValueGroupWithZero.mk r s)

Tactic:
simp [Valuation.IsEquiv.orderMonoidIso_spec₀]

Goal state after:
Module: Mathlib.Topology.Algebra.UniformField

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
⊢ ∀ ⦃x : hat K⦄, x ≠ 0 → ContinuousAt Inv.inv x

Tactic:
intro x

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
x : hat K
⊢ x ≠ 0 → ContinuousAt Inv.inv x
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R : Type u_1
Γ₀ : Type u_2
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : Ring R
v : Valuation R Γ₀
S : Type u_3
inst✝¹ : Ring S
Λ₀ : Type u_4
inst✝ : LinearOrderedCommGroupWithZero Λ₀
w : Valuation S Λ₀
s : Set (WithVal v)
x : WithVal v
e : ValuativeRel.ValueGroupWithZero (WithVal v) ≃*o (MonoidWithZeroHom.ofClass (valuation v)).ValueGroup₀ :=
  ValuativeRel.ValueGroupWithZero.orderMonoidIso (valuation v)
⊢ (∃ γ, {y | Valued.v.restrict (y - x) < ↑γ} ⊆ s) ↔ ∃ γ, {a | (ValuativeRel.valuation (WithVal v)) (-x + a) < ↑γ} ⊆ s

Tactic:
apply e.unitsCongr.symm.exists_congr fun a ↦ ?_

Goal state after:
R : Type u_1
Γ₀ : Type u_2
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : Ring R
v : Valuation R Γ₀
S : Type u_3
inst✝¹ : Ring S
Λ₀ : Type u_4
inst✝ : LinearOrderedCommGroupWithZero Λ₀
w : Valuation S Λ₀
s : Set (WithVal v)
x : WithVal v
e : ValuativeRel.ValueGroupWithZero (WithVal v) ≃*o (MonoidWithZeroHom.ofClass (valuation v)).ValueGroup₀ :=
  ValuativeRel.ValueGroupWithZero.orderMonoidIso (valuation v)
a : (MonoidWithZeroHom.ofClass (valuation v)).ValueGroup₀ˣ
⊢ {y | Valued.v.restrict (y - x) < ↑a} ⊆ s ↔
    {a_1 | (ValuativeRel.valuation (WithVal v)) (-x + a_1) < ↑(e.unitsCongr.symm.toEquiv a)} ⊆ s
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: case IJ
Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
π : v.Uniformizer
⊢ maximalIdeal ↥v.valuationSubring ≤ span {π.val}

Tactic:
intro x

Goal state after:
case IJ
Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
π : v.Uniformizer
x : ↥v.valuationSubring
⊢ x ∈ maximalIdeal ↥v.valuationSubring → x ∈ span {π.val}
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: case h
R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : v.IsEquiv w
γ : (ofClass Valued.v).ValueGroup₀ˣ
r s : R
hr₀ : 0 < Valued.v r
hs₀ : 0 < Valued.v s
hr : Valued.v.restrict r / Valued.v.restrict s = ↑γ
⊢ ∀
    x ∈
      {x |
        Valued.v.restrict x <
          ↑(Units.mk0 (Valued.v.restrict ((equiv v).symm r) / Valued.v.restrict ((equiv v).symm s)) ⋯)},
    (equiv v) x ∈ {x | Valued.v.restrict x < ↑γ}

Tactic:
intro x

Goal state after:
case h
R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : v.IsEquiv w
γ : (ofClass Valued.v).ValueGroup₀ˣ
r s : R
hr₀ : 0 < Valued.v r
hs₀ : 0 < Valued.v s
hr : Valued.v.restrict r / Valued.v.restrict s = ↑γ
x : WithVal v
⊢ x ∈
      {x |
        Valued.v.restrict x <
          ↑(Units.mk0 (Valued.v.restrict ((equiv v).symm r) / Valued.v.restrict ((equiv v).symm s)) ⋯)} →
    (equiv v) x ∈ {x | Valued.v.restrict x < ↑γ}
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: case h
R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : w.IsEquiv v
γ : (ofClass Valued.v).ValueGroup₀ˣ
r s : WithVal v
hr₀ : 0 < Valued.v r
hs₀ : 0 < Valued.v s
hr : Valued.v.restrict r / Valued.v.restrict s = ↑γ
h' : w.restrict.IsEquiv v.restrict
⊢ ∀ x ∈ {x | Valued.v.restrict x < ↑(Units.mk0 (Valued.v.restrict ((equiv v) r) / Valued.v.restrict ((equiv v) s)) ⋯)},
    (equiv v).symm x ∈ {x | Valued.v.restrict x < ↑γ}

Tactic:
intro x

Goal state after:
case h
R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : w.IsEquiv v
γ : (ofClass Valued.v).ValueGroup₀ˣ
r s : WithVal v
hr₀ : 0 < Valued.v r
hs₀ : 0 < Valued.v s
hr : Valued.v.restrict r / Valued.v.restrict s = ↑γ
h' : w.restrict.IsEquiv v.restrict
x : R
⊢ x ∈ {x | Valued.v.restrict x < ↑(Units.mk0 (Valued.v.restrict ((equiv v) r) / Valued.v.restrict ((equiv v) s)) ⋯)} →
    (equiv v).symm x ∈ {x | Valued.v.restrict x < ↑γ}
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
h : v.IsEquiv w
h_val : Valued.v.IsEquiv Valued.v
h_res : v.restrict.IsEquiv w.restrict
⊢ ∀ (ib : (ofClass Valued.v).ValueGroup₀ˣ),
    ∃ ia, ∀ x ∈ {x | Valued.v.restrict x < ↑ia}, (RingHom.id R) x ∈ {x | Valued.v.restrict x < ↑ib}

Tactic:
intro x

Goal state after:
R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
h : v.IsEquiv w
h_val : Valued.v.IsEquiv Valued.v
h_res : v.restrict.IsEquiv w.restrict
x : (ofClass Valued.v).ValueGroup₀ˣ
⊢ ∃ ia, ∀ x_1 ∈ {x | Valued.v.restrict x < ↑ia}, (RingHom.id R) x_1 ∈ {x_2 | Valued.v.restrict x_2 < ↑x}
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: K : Type u_7
inst✝¹ : DivisionRing K
Γ₀ : Type u_8
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
hv : Function.Surjective ⇑v
γ : Γ₀ˣ
⊢ ∃ r s, 0 < v r ∧ 0 < v s ∧ v r / v s = ↑γ

Tactic:
obtain ⟨r, hr⟩ := hv γ
  exact ⟨r, 1, by simp [hr]⟩

Goal state after:
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
⊢ ∀ (a : X), ↑a ∈ s → ∀ (a_1 : X), ↑a_1 ∈ s → a < a_1 → ∃ a_3, (↑a_3 ∈ s ∧ a < a_3) ∧ a_3 < a_1

Tactic:
intro x

Goal state after:
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
x : X
⊢ ↑x ∈ s → ∀ (a : X), ↑a ∈ s → x < a → ∃ a_2, (↑a_2 ∈ s ∧ x < a_2) ∧ a_2 < a
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
n : ℕ
u : Aˣ
ha : ¬↑u * π ^ n = 0
m : ℕ
w : Aˣ
h_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A K) (↑w * π ^ m) = x
hb : π ^ m ∈ nonZeroDivisors A
H : (maximalIdeal A).intValuation π ^ n / (maximalIdeal A).intValuation π ^ m ≤ 1
h_mn : m ≤ n
⊢ π ∈ nonZeroDivisors A

Tactic:
rw [mem_nonZeroDivisors_iff_ne_zero]

Goal state after:
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
n : ℕ
u : Aˣ
ha : ¬↑u * π ^ n = 0
m : ℕ
w : Aˣ
h_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A K) (↑w * π ^ m) = x
hb : π ^ m ∈ nonZeroDivisors A
H : (maximalIdeal A).intValuation π ^ n / (maximalIdeal A).intValuation π ^ m ≤ 1
h_mn : m ≤ n
⊢ π ≠ 0
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
⊢ ∀ (x : K), x ≠ 0 → ∃ U ∈ 𝓝 0, x ∉ U

Tactic:
intro x

Goal state after:
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
x : K
⊢ x ≠ 0 → ∃ U ∈ 𝓝 0, x ∉ U
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ ∀ (x : K), ContinuousAt (⇑v.restrict) x

Tactic:
intro x

Goal state after:
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : K
⊢ ContinuousAt (⇑v.restrict) x
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
⊢ ∀ (x : K), ContinuousAt (⇑v) x

Tactic:
intro x

Goal state after:
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
x : K
⊢ ContinuousAt (⇑v) x